Segal K-theory of vector spaces with an automorphism

We describe the Segal $K$-theory of the symmetric monoidal category of finite-dimensional vector spaces over a perfect field $\mathbb{F}$ together with an automorphism, or, equivalently, the group-completion of the $E_\infty$-algebra of maps from $S^1$ to the disjoint union of classifying spaces $\mathrm{BGL}_d(\mathbb F)$, in terms of the $K$-theory of finite field extensions of $\mathbb{F}$. A key ingredient for this is a computation of the Segal $K$-theory of the category of finite-dimensional vector spaces with a nilpotent endomorphism, which we do over any field $\mathbb F$. We also discuss the topological cases of $\mathbb F =\mathbb C,\mathbb R$.